Dan Liu

Xiangjian He

Yicheng Jin

CiteSeerX

Proceedings of the 2nd International Conference on Information Technology for Application (leITA 2004)
The Bivariable Fractal Interpolation Algorithm
of Simulating the Mountains in the Distributed
Navigation Simulation System
Dan Liu, Xiangjian He, and Yicheng Jin
Ah.·'ifru('I--3D scene system is an important part of computer
imitation system. and its fidelity determines if an imitation system
stands or falls. At present. such algorithms han been perfected,
but a good algorithm is always so complicated that can not achieve
the demands of calculation during the time required. So it is
necessary to find a fast algorithm applied in the real time system.
Fractal Geometry is a PO\\ ertul tool to describe the complicated
and anomalistic geometrical objects. A method of bivariable
fractal interpolation combined with objects polyhedral technique
to construct polyhedrons and vertex data of mountains in
distributed simulation system is proposed. The implementing
method and steps an.' given as well.
Index Terms-Bivariable fractal function, Bivariable fractal
interpolation, Distributed navigation simulation system,
Mouutains simulating.
!:"'TRODL:C no-,
Computer simulation technique is a multi-subject synthesistechnique based on system techniques. information
techniques and related speciality techniques in the
application fields. It uses computers and various physical
domino affecting facilities and dynamically studies actual
systems or systems in design by system models. Moreover.
distributed navigation simulation system is a large distributed
interactive computer simulation system combining virtual
reality techniques, internet techniques, multimedia techniques
and navigation techniques. which has been applied broadly in
the fields such as system analysis, scientific research and
engineering design. As an important part of the system. 3D
scene system with accompanying sound system and interactive
equipment makes a virtual reality environment with an
experiential feeling for the users il] .
Since ships are alwavs navigating. the mountains on the
seacoast comprise the main contents of the navigation
simulation system. Natural objects such as mountains are very
anornalistic. These objects can only be approximated by
polyhedrons in computer graphics. which is an effective
Dan Liu is with Institute of Nautical Science&T cchnology. Dnlian Maritime
Universirv. Dahan. Liuoninu. 1160~6 P.R.China (e-mail: dliu dlnur« 263.nct)
Xiangjiun He is with ~Department of Computer Svsterns. Faculty of
Information Technologv t'nivcrsiry of Technology. Svdney. PO Box 123.
Broadway ~007 .. Australia t c-muil sean-«. it.uts.cdu.aui.
Yicheng Jin IS with Institute of Nautical Scicncce.Technologv. Dulian
'vlaritimc Univerotv. Dahan. Liaonine. 11602(1 P.RThin,] {e-mail:
dhu dlmu([! 263.nel). ~
ICITA2004 IS8:Y 0-646-423/3-4
figuration method of the geometrical objects with certain
lubricous degree and leads to more errors in describing natural
objects. Mountains within a certain range can be seen as a
compact set of R' Space and fractal interpolation method is an
effective \\'ay to construct Iterated Function System (IFS) using
a group of existing data. we will discuss how to collect original
data and usc these data to construct lFS in detail in this paper:
Fractal geometry obtains the attractor through infinite
alternation. which can not be used by a real time system.
Consequently some useful results of bivariable functions have
been obtained by studying the hivariable fractal space and how
to use bivariablc fractal method to produce the basic data in the
polyhedron presentation method. This is followed by a
discussion of how to take full advantage of the validity of the
fractal method and efficiency of the polyhedron presentation
method are abo be discussed.
II. THE COLLECTtO,," .-\'<DPRETRE.·I'T\tE'oJT OF ORtGtNAL O.\L\
A. The Coll ection ofOriginal Data
The original data here is defined as a discrete point set
obtained from data source. At present. there are three
approaches to obtain the original data: using digital instruments
to collect needed information from the chart papers; using a
mouse to pick up information from raster charts; or using
special software to pick up data automatically under operator's
commands. From the contour lines ofa mountain we can figure
the detailed shape of the mountain. It is reasonable and feasible
to get the characteristic information of reconstructing the
mountain's shape. The method is to collect the 3D coordinates
of a series points on the contour lines dispersedly. In fact. the
planar coordinates can be obtained easily by tracking the
location of digital collector or cursor. In order to gain the third
coordinate. which is the height information. it is required to
input the altitude value of current contour line as well when
collecting the data. In general. to ensure the obtained
information can express the mountain information effectively.
it is requested that the sampling frequency is higher when the
curvature of the contour line is bigger. and is lower when the
curvature is smaller in order to reduce the number of data.
B. The Preprocessing otthe Messv Data
The collected data can be used as interpolating data only
after a series of pretreatments. The pretreatments include the
coordinate transforrnation between different coordinates and
185
Proceedings of the 2nd International Conference on Information Technology for Application (ICITA 20()~)
the sort of the coordinate values. It can be seen as follows that
both treatments are in the interest of satisfying the fractal
interpolation methods with the interpolating data. In the
following. we will take the example of collection by digital
instrument to introduce the special process of data
preprocessing.
First of all, we will introduce three coordinate systems used
here; they are facility coordinate system. geographical
coordinate system and world coordinate system. Facility
coordinate system is the coordinate system of digital instrument
itself. The origin is the lower-left corner of the digital
instrument; right is the positive direction of X-Axis and up is
the positive direction ofY -Ax is, and the coordinate unit is pixel.
Secondly. in order to simplify the problem. we introduce a third
coordinate, measured in meters to express the height
information of the point; Geographical coordinate system uses
latitude and longitude to express the coordinate value of a point.
We also introduce a third coordinate; World coordinate system
is the left hand coordinate system used in the 3D scene system,
the origin is the low-left corner of the chart, and the coordinate
unit is meter. Since there is no need to change the height
information in the course of transformation. we do not consider
transforming the height value. In this case. we simplify the
transformation between three-dimensional coordinate systems
to the transformation between two-dimensional coordinate
systems. The three plane coordinate system are X- Y plane of
facility coordinate system, geographical coordinate plane of
geographical coordinate system and X-Z plane of world
coordinate system. and the height value (Z value) is
transformed to the Y value in the world coordinate system
directly.
After the coordinate of each point is transformed into world
coordinate system. it will be sorted in X-Z plane. The sort is
going along two directions; one is sorting every point
ascendingly according to its X value along X direction. The
points with same value can only be used once. In this way, there
is a partition on the X-Axis composed of X value of each point:
{[x", xl],[xl,x2], .... [xM_I,xM]I, M+! is the number of points
with different coordinate value, Xmin= x., Xm,,= x". It can be
imagined that this partition is nonsymmetricaL In the same way,
we can gain a partition along Z-Axis: {[z",
zd,[ZJ,Z2]," .. [z,.J,z,]l by sorting interval [Zmm'zm,,]. where, N
has the same meaning with M.
C Computing Output Data
When IFS is gained, we can iterate an arbitrary initial set to
obtain the attractor of the IFS. which is the final output result.
In order to obtain the graph with smaller error, it usually needs
additional iteration. Such iteration process sometimes takes
time, especially when there are more mappings in the IFS.
Moreover, in a real time simulation application. in order to gain
better visual effect. the updating rate should be greater than 20
frames per second. Thereby, it is infeasible to gain a frame
image using several seconds or more by iteration. Current
graphics softwares can support the protraction of polyhedrons
better and the efficiency is higher. That is the reason that most
applications usc polyhedrons to denote geometrical object. So
in order to combine fractal interpolation method with practical
application, it is necessary to combine the method with
traditional polyhedron protraction method. This method can
ensure not only higher precision of the graphics obtained but
also abundant details. We do it as follows: firstly. use IFS to
compute offline. after a few iterations. abandon the middle
result. and let the last data be the output result. Secondly,
construct polyhedrons grid using this output discrete data by
messy data interpolation method. In the course of the iteration,
in order to make the iteration convergent to a given precision,
usually use the point in the attractor as the original iteration
point. In this case, we can get a satisfying result after 3 to 5
iterations. The method of constructing polyhedrons grid can
select Delaunay triangular grid automatic generation algorithm
in common use.
III. THE BIVARIABLE FRACTAL INTERPOLATION FUNCTION
Let X be the complete metric space and let W j :X--->X
;=J,2, .."n be continuous mappings. Then the collection :x.-
W j :i=!,2, ... .n} is called an iterated function system,
abbreviated IFS. A set A <:;;; X is said to be an attractor of the
n
IFS if A is nonernpty, compact and with A=UlY, (A).We
i-==l
now introduce some notations which we will put to use
throughout the rest of this paper [21.
Let A. E (-1.1) be a constant, m.n ~ 12 be integers. Denote
M = : I, ....m I and N = ( I, ....n J .
Let - XJ ::; Xo < XI < ... < xm < XJ and
- XJ < Yo < YI < ... < y" < XJ Denote D=[ x"' xm ] x
[Yo.y,,] and u., =[X,_IXJX[Y,_I'Y,] for all iEMand
j E N. Let qD) denote the collection of real-valued continuous
functions defined on D. The partition of D given by
D= U Dj.! will be denoted by t,. For; EM and j EN define
i.j
maps Ai :[xO,xm]~[-\',_I,-'J
8, : [Yo,Y,,] ~ LvH,y,] by
and
X-X. I
A,(x) =-'--'--(x-x,,) +xj .I'X E [XO,Xm]s; -xo
and
respectively.
Definition l.Let rv,., : D x R ~ D x R, (i, j) E M x 1'1,
be continuous mappings. The IFS
: D x R; rv,.; : (i, j) E M x 1'1 J is said to be iterated
generating system, called IGS[J[ for short, if there is a function
f E C(D) such that ((x, y, f (x, y)):(x. y) ED:. the graph of
f. is the unique attractor of the IFS, in which case we also
186
write.
Proceedings of the 2nd Intemational Conference on Information Technology for Application (ICIT A 2004)
IV. THE SPACE OF BIVARIABLE FRACTAL INTERPOLATION
FUNCTIONS
Theorem 1 1~1 .Let ¢i;EC(D),(i,})EMxN be
Lipschitz. Define Wi./ : Dx R ~ Dx R by
w,.;(x,y, z) = (Ai(x), B/(y),k + ¢i)X,y))
For (x, y, z) E Dx Rand (i, j) E M x N. then
:D x R; w,; :(i,}) E M x N : is an IGS ifand only if there
exist lIEC[X",.l,,,] and VEC[Yo,yJ
following equations hold:
AU(x) = ¢i/(X,y")-¢iH(X'YO)
x E [Xu,xn,], i E M,j = l,.",n-1
AV(X) = ¢i)Xm, y) - ¢i+l.,(-'-'U'y)
V. CONCLUSION
Theorem 3. Suppose A * ° , Let J be a subset of
(i.j): i = O,l"",m } = O.1,· .. .n :.Denote
Jf =JUI(L1) and r, =JUB(L1).Let :Zil :(i,})EJ I
be a set of real numbers. Then we have (I). There exists an
IE F(L1, A, K) such that
I(xi.y,) = Zi;,(i,}) E J
If and only if the following equations
(1)
such that the +(xi.I-.\·,,)(zm.O-zm ..)=0,i=2"",m
(Yn - Y H )(zo.o - zm.u) - (v" - Yo )(ZO./.I -
Y E Lv",y,,],) EN,} = 1,. .. ,m-1
11(Ai (x)) = AU(X) + «: (x, Yo) -¢i" (x, Y,,)
X E [x",xm],i E M
v(B/v)) = AV(Y) + ¢IJ'-'uY) -¢m,(Xm'Y)
Y E [y",y,,],} EN
Definition 2 Let H be a linear subspace ofC (D) then the set
F(L1, A, H) =
I IEC(D): there exist ¢i.; EH,(i,})E MxN, such
that
DxR;W,.;:(i,})EMxN} <=> I
w"j: DxR ~ Dx R is defined by
w"j (x, y, z)=( Ai (x), B; (v), k + ¢ij (x, Y, z»}
is called the set offractal functions with respect to the partition
L1,the contractive factor A and the bottom space H.
Theorem 2.Denote K~ (¢ E C(D): there exist a, b. c and d in
R such that ¢ (x, y)= axy+ bx + cy + d for all (x, v) ED)
Then F ( L1. A , K) is a linear subspace of C (D). Furthermore,
we have
{
mn + 3 ijA * °
dim F( L1, A ,K)=
(m+1)(n+1) ijA=O
Let us denote
l(L1) = {(i,}): i = I," ',m -I;} = 1, .. · ,n-I}
Called the interior indices, and
B(L1) = {(i,}): i = O,m;} = O,l, .. ·,n} U {(i,}):
i = O,l, .. ·,m:} = O,n}
Called the boundary indices.
2",.,._1) + ev i-I - Yo )(ZO.n - Zm.n) = 0 (3)
where
} = 2,.··, n
In the unknowns Zi./ (i,}) E B(L1) \ J e .are linearly solvable.
There exists one and only IE F(L1, A, K) such that (1)
holds if and only if Jf = l(L1) and the equation (2) and (3) in the
unknowns Zi.; (i,}) E B(L1) \ J s (see [2]), are linearly solvable.
Corollary. Suppose A * ° and let
J={(i,j): i= O,l,.··,m -1;
} = 0,1, .. · ,n-l} U {(O.nl(III,OUIII.n)}
Then for any set of mn+ 3 real numbers I z,., : (i,}) E J l , there
exists a unique IE F(L1,A, K) such that (1 )holds. Furthermore.
for each ( p. q ) E J , let gp,q E F (L1,A, K) be the unique
{
I i(i,})=(p,q)
function with g p." (Xi' Y;) = °
i(i,}) E J \ {(p,q)}
Then {gi.; : (i,}) E J } forms a basis of the linear space
F(L1,A,K).
According to Theorem 3 and the Corollary, we can
outline mountain in the screen plane of computer. Figure 1 is
the experimental result of this method.
187
Proceedings of the 2nd International Conference on Information Technology for Application (ICIT A 2004)
Fig. I. The 3D 11100et of three hill island. Dulian, China
In fact. common ship often navigates along the
recommended sea-route when it navigates along the seashore.
The outline of a mountain can be seen to be composed of the
points of intersection between the normal of the center of the
sea-route and the contour line with the most height and the
points with the nearest distance to the sea-route center when the
measurer observes the mountain in the positive horizontal
direction. Therefore, the means to pick up the outline points of
the mountain is: firstly, separate the center line of the sea-route
with equal intervals; secondly, fine the intervals. The unequal
interval partition can be used according to such factors as the
ridge lines, the changes of height value and the curvature of the
sea-route center line. Rearrange the sampling points in a certain
order, and then put them into the computer. The following
figure is the mountain outline of Huangbaizui, Dalian.
Fig. ') The mountain outline of l luangbaizui, Dalian. China
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The Bivariable Fractal Interpolation Algorithm of Simulating the Mountains in the Distributed Navigation Simulation System

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The Bivariable Fractal Interpolation Algorithm of Simulating the Mountains in the Distributed Navigation Simulation System

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